Isométries vectorielles

f∈ O(E)V E
V f V ⊥
E E
V E
f(V⊥)⊂(f(V))⊥?
f∈ O(E)f
E f E
∀x, y ∈E, (x|y) = 0 =⇒(f(x)|f(y)) = 0
(u+v|u−v)u, v
α∈R+
∀x∈E, 
f(x) =α x
g∈ O(E)f=α.g
E f ∈ L(E)
f
f2=−Id
f(x)x x
n∈N∗M Mn(R)
ϕ: (A, B)∈ M27→ tr tAB
ϕ
Ω∈ M M7→ ΩM
ϕ
(E, h., .i)
ϕ∈ O(E)M(ϕ) = Im(ϕ−IdE)F(ϕ) = ker(ϕ−IdE)
u∈E\ {0}suu⊥
ϕ∈ O(E)M(ϕ)⊕⊥F(ϕ) = E
(u1, . . . , uk)
M(su1◦ · · · ◦ suk) = Vect(u1, . . . , uk)
(u1, . . . , uk)v1, . . . , vk∈E\ {0}
su1◦ · · · ◦ suk=sv1◦ · · · ◦ svk
(v1, . . . , vk)
(.|.)Rn
u= (u1, . . . , up)∈(Rn)p
Mu= ((ui|uj))1≤i,j≤p
(u1,...up)Mu
u= (u1, . . . , up)v= (v1, . . . , vp)
Mu=Mv
f∈ O(Rn)f(ui) = f(vi)i
u E v =u−IdE
ker v= (Im v)⊥

un=1
n
n−1
X
k=0
uk
un(x)n∈Nx
xker v
u E n
v=u−Id
ker v= (Im v)⊥
x∈E(x1, y)∈ker v×E
x=x1+v(y)
1
N
N−1
X
k=0
uk(x) = x1+1
N(uN(y)−y)
pker v
∀x∈E, lim
N→+∞
p(x)−1
N
N−1
X
k=0
uk(x) = 0
p, q A, B Mp,q(R)
tAA =tBB
ker Aker B
f g RqRpA
BRqRpRp
∀x∈Rq,hf(x), f(y)i=hg(x), g(y)i
(ε1, . . . , εr) (ε0
1, . . . , ε0
r)F
r
∀(i, j)∈ {1, . . . , r}2,hεi, εji=ε0
i, ε0 j
s F
∀i∈ {1, . . . , r}, s(εi) = ε0
i
U∈ Op(R)A=UB
EO(E)
E
Γ = {u∈ L(E)| ∀x∈E, ku(x)k≤kxk}
ΓL(E)O(E)
u∈Γ (f, g)∈Γ2
f6=g u =1
2(f+g)
u /∈ O(E)
v E ρ ∈ O(E)s
E v =ρ◦s
v
u∈Γ
(f, g)∈Γ2
f6=g u =1
2(f+g)

=⇒V f f(V)⊂V f
f(V) = V x ∈V⊥y∈V
(f(x)|y)=(x|f−1(y)) = 0
f−1(y)∈V f(x)∈V⊥V⊥f
⇐=V⊥f V =V⊥⊥
x, y
kxk=kykx+y x −y f(x) + f(y)f(x)−f(y)
kf(x)k=kf(y)k
(e1, . . . , en)
f=λg g ∈ O(E)
λ f x f(x) = λx
kf(x)k=kxkλ=±1f
−1
(u+v|u−v) = kuk2− kvk2= 0 u v
u v E
u+v u −v f(u+v)f(u−v)
f(u+v) = f(u) + f(v)f(u−v) = f(u)−f(v)
kf(u)k=kf(v)k
E f
α∈R+
∀x∈E, kf(x)k=αkxk
x∈E
x= 0 f(x)=0 kf(x)k=αkxk
x6= 0 u=x/kxk kf(u)k=α
kf(x)k=αkxk
α= 0 f=˜ 0g∈ O(E)
α6= 0
g=1
αf
g g ∈ O(E)
x∈E
(f(x)|x) = −f(x)|f2(x)=−(x|f(x))
(f(x)|x)=0
x+f(x)f
(x+f(x)|f(x+f(x))) = (x+f(x)|f(x)−x)=0
kf(x)k2=kxk2f
x y
f2(x) + x|f(y)= (f(x)|y)+(x|f(y))
(f(x+y)|x+y)=(f(x)|y)+(f(y)|x)=0
f2(x) + x|f(y)= 0
f f2(x) + x= 0E

Mn(R)
f:M7→ ΩM(f(M)|f(N)) = tr(tMtΩΩN)
f ϕ M, N ∈ M
(M|tΩΩN)=(M|N)N∈ M tΩΩN=NtΩΩ = In
f ϕ Ω
y∈M(ϕ)x∈F(ϕ)
ϕ(x) = x a ∈E y =ϕ(a)−a
hx, yi=hx, ϕ(a)i−hx, ai=hϕ(x), ϕ(a)i−hx, ai= 0
ϕ∈ O(E)
M(ϕ)F(ϕ)
dim M(ϕ) + dim F(ϕ) = dim E
M(ϕ)⊕⊥F(ϕ) = E
k≥1
k= 1
k≥1
(u1, . . . , uk+1)ϕ=su1◦ · · · ◦ suk+1 ∈ O(E)
F(ϕ)
x∈F(ϕ)ϕ(x) = x
su1◦ · · · ◦ suk(x) = suk+1 (x)
su1◦ · · · ◦ suk(x)−x=suk+1 (x)−x
suk+1 (x)−x∈Vect(uk+1)
su1◦ · · · ◦ suk(x)−x∈Vect(u1, . . . , uk)
(u1, . . . , uk+1)
su1◦ · · · ◦ suk(x)−x=suk+1 (x)−x= 0
x su1◦ · · · ◦ suksuk
x∈Vect(u1, . . . , uk)⊥∩Vect(uk+1)⊥= Vect(u1, . . . , uk+1)⊥
F(ϕ)⊂Vect(u1, . . . , uk+1)⊥
F(ϕ) = Vect(u1, . . . , uk+1)⊥
M(ϕ) = Vect(u1, . . . , uk+1)
ϕ=su1◦ · · · ◦ suk=sv1◦ · · · ◦ svk
F(ϕ) = Vect(u1, . . . , uk)⊥
Vect(v1, . . . , vk)⊥⊂F(ϕ)
Vect(u1, . . . , uk)⊂Vect(v1, . . . , vk)
(u1, . . . , uk)
(v1, . . . , vk)
C1, . . . , CpMu
(u1, . . . , up)λ1, . . . , λp
λ1u1+· · · +λpup= 0E
∀1≤i≤p, (λ1u1+· · · +λpup|ui)=0
λ1C1+· · · +λpCp= 0
Mu
Mu
λ1, . . . , λp
λ1C1+· · · +λpCp= 0

∀1≤i≤p, (λ1u1+· · · +λpup|ui)=0
λ1u1+· · · +λpup∈Vect(u1, . . . , up)⊥
λ1u1+· · · +λpup∈Vect(u1, . . . , up)
λ1u1+· · · +λpup= 0E
(u1, . . . , up)
r= rg(u1, . . . , up) (u1, . . . , up)
r u
(v1, . . . , vp)
Mu=Mvr
v
h: Vect(u1, . . . , ur)→Vect(v1, . . . , vr)
∀1≤k≤r, h(uk) = vk
x=λ1u1+· · · +λrur
h(x) = λ1v1+· · · +λrvr
kxk2=
r
X
i,j=1
λiλj(ui|uj)kh(x)k2=
r
X
i,j=1
λiλj(vi|vj)
(ui|uj)=(vi|vj)
kxk2=kh(x)k2
h
k∈ {r+ 1, . . . , p}uku1, . . . , ur
uk=λ1u1+· · · +λrur
i∈ {1, . . . , r}
(uk−(λ1u1+· · · +λrur)|ui)=0
(vk−(λ1v1+· · · +λrvr)|vi)=0
vk=λ1v1+· · · +λpvrvk=h(uk)
hRn
Vect(u1, . . . , ur)⊥Vect(v1, . . . , vr)⊥
x∈ker v y =v(a)∈Im v u(x) = x y =u(a)−a
(x|y)=(u(x)|u(a)) −(x|a)=0
uker v⊂(Im v)⊥
x∈E x =a+b a ∈ker v b ∈(ker v)⊥= Im v
u(a) = a uk(a) = a k ∈Nc
b=v(c) = u(c)−c uk(b) = uk+1(c)−uk(c)
un(x) = a+1
nun(c)−1 nc
u
1
nun(c)
=1
nkck → 0
un(x)→a
x∈ker v y =v(a)∈Im v u(x) = x y =u(a)−a
(x|y)=(u(x)|u(a)) −(x|a)=0
u
ker v⊂(Im v)⊥
E= ker v⊕⊥Im v
x1y
k∈N
uk(x1) = x1uk(v(y)) = uk+1(y)−uk(y)
1
N
N−1
X
k=0
uk(x) = x1+1
N(uN(y)−y)